### One Factor Design: An Introduction

A one factor design (also known as a one-way ANOVA) is a statistical method used to determine if there are significant differences between the means of multiple groups. In this design, there is one independent variable (factor) with multiple levels or categories.

## Table of Contents

Suppose $y_{ij}$ is the response is the $i$th treatment for the $j$th experimental unit, where $i=1,2,\cdots, I$. The statistical model for a completely randomized one-factor design that leads to a One-Way ANOVA is

$$y_{ij} = \mu_i + e_{ij}$$

where $\mu_i$ is the unknown (population) mean for all potential responses to the $i$th treatment, and $e_{ij}$ is the error (deviation of the response from population mean).

The responses within and across treatments are assumed to be independent and normally distributed random variables with constant variance.

### One Factor Design’s Statistical Model

Let $\mu = \frac{1}{I} \sum \limits_{i} \mu_i$ be the grand mean or average of the population means. Let $\alpha_i=\mu_i-\mu$ be the $i$th group treatment effect. The treatment effects are constrained to add to zero ($\alpha_1+\alpha_2+\cdots+\alpha_I=0$) and measure the difference between the treatment population means and the grand mean.

Therefore the one way ANOVA model is $$y{ij} = \mu + \alpha_i + e_{ij}$$

$$Response = \text{Grand Mean} + \text{Treatment Effect} + \text{Residuals}$$

From this model, the hypothesis of interest is whether the population means are equal:

$$H_0:\mu_1=\mu_2= \cdots = \mu_I$$

The hypothesis is equivalent to $H_0:\alpha_1 = \alpha_2 =\cdots = \alpha_I=0$. If $H_0$ is true, then the one-way ANOVA model is

$$ y_{ij} = \mu + e_{ij}$$ where $\mu$ is the common population mean.

### One Factor Design Example

Let’s say you want to compare the average test scores of students from three different teaching methods (Method $A$, Method $B$, and Method $C$).

**Independent variable:**Teaching method (with three levels: $A, B, C$)**Dependent variable:**Test scores

### When to Use a One Factor Design

**Comparing means of multiple groups:**When one wants to determine if there are significant differences in the mean of a dependent variable across different groups or levels of a factor.**Exploring the effect of a categorical variable:**When one wants to investigate how a categorical variable influences a continuous outcome.

### Assumptions of One-Factor ANOVA

**Normality:**The data within each group should be normally distributed.**Homogeneity of variance (Equality of Variances):**The variances of the populations from which the samples are drawn should be equal.**Independence:**The observations within each group should be independent of each other.

### When to Use One Factor Design

- When one wants to compare the means of multiple groups.
- When the independent variable has at least three levels.
- When the dependent variable is continuous (e.g., numerical).

Note that

If The Null hypothesis is rejected, one can perform post-hoc tests (for example, Tukey’s HSD, Bonferroni) to determine which specific groups differ significantly from each other.

**Remember**: While one-factor designs are useful for comparing multiple groups, they cannot establish causation.